Tokio Matsuyama

Global well-posedness of Kirchhoff systems☆

Abstract The aim of this paper is to establish the H 1 global well-posedness for Kirchhoff systems. The new approach to the construction of solutions is based on the asymptotic integrations for strictly hyperbolic systems with time-dependent coefficients. These integrations play an important role to setting the subsequent fixed point argument. The existence of solutions for less regular data is discussed, and several examples and applications are given.

Asymptotics for the nonlinear dissipative wave equation

We are interested in the asymptotic behaviour of global classical solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like α 1 (1+|x|) -δ |u t | β u t (α 1 , β, δ > 0) and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as the time goes to infinity.

Scattering for strictly hyperbolic systems with time‐dependent coefficients

The present paper is devoted to finding a necessary and sufficient condition on the occurence of scattering for the regularly hyperbolic systems with time-dependent coefficients whose time-derivatives are integrable over the real line. More precisely, it will be shown that the solutions are asymptotically free if the coefficients are stable in the sense of the Riemann integrability as time goes to infinity, while each nontrivial solution is never asymptotically free provided that the coefficients are not R-stable as times goes to infinity. As a by-product, the scattering operator can be constructed. It is expected that the results obtained in the present paper would be brought into the study of the asymptotic behaviour of Kirchhoff systems.

Global Well-Posedness of the Kirchhoff Equation and Kirchhoff Systems

This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.

A necessary and sufficient condition on scattering for the regularly hyperbolic systems

The present paper is devoted to finding a necessary and sufficient condition on the occurence of scattering for the regularly hyperbolic systems with time-dependent coefficients whose time-derivatives are integrable over the real line. More precisely, it will be shown that the solutions are asymptotically free if the coefficients are stable in the sense of the Riemann integrability as time goes to infinity, while each nontrivial solution is never asymptotically free provided that the coefficients are not R-stable as times goes to infinity. As a by-product, the scattering operator can be constructed. It is expected that the results obtained in the present paper would be brought into the study of the asymptotic behaviour of Kirchhoff systems.

RAPIDLY DECREASING SOLUTIONS AND NONRELATIVISTIC LIMIT OF SEMILINEAR DIRAC EQUATIONS

In this paper we study the existence of the solutions for the nonlinear Dirac equations in the weighted Sobolev spaces and the Schwartz space. It is also shown that these solutions converge to the solutions for the nonlinear Schrodinger type equations if the speed of light is regarded as infinity.

Lp-boundedness of Functions of Schrödinger Operators on an Open Set of \(\mathbb{R}^{d}\)

The purpose of this paper is to prove L p -boundedness of an operator φ(H V ), where \(H_{V } = -\Delta + V (x)\) is the Schrodinger operator on an open set \(\Omega \) of \(\mathbb{R}^{d}\) (d ≥ 1). Moreover, we prove uniform L p -estimates for φ(θ H V ) with respect to a parameter θ > 0. This paper will give an improvement of our previous paper (Iwabuchi et al., L p -mapping properties for Schrodinger operators in open sets of \(\mathbb{R}^{d}\), submitted); assumptions of potential V and space dimension.

The Kirchhoff Equation with Gevrey Data

In this article the Cauchy problem for the Kirchhoff equation is considered, and the almost global existence of Gevrey space solutions is described.

Perturbed Besov Spaces by a Short-range Type Potential in an Exterior Domain

The aim of this work is to provide the equivalence relation between perturbed Besov spaces by a short-range potential in an exterior domain outside a convex obstacle, and the free ones.

Representation Formula of the Resolvent for Wave Equation with a Potential Outside the Convex Obstacle

The purpose of the present article is to provide the results on the perturbed resolvent of the Schrodinger operator in an exterior domain outside the convex obstacle. The representation formula for this resolvent via the free one in the whole space\( \mathbb{R}^n \) will be given, which extends the result of [5] to higher-dimensional spaces.